Why are these calibers used in the military? 7.62, 9 and 5.56?

When I was a draftee in the late 90s in the German Bundeswehr, the HK G3 was the standard rifle with its 7.62mm ammunition. We also fired the Uzi with its 9mm ammo. As a sidearm we had the Walter P1, also using the 9mm ammo.

The AK-47 is famous for the 9mm caliber.

Then - at the end of my service - introduced rifle HK G36 uses the now more common (at least from my layman's perspective) 5.56mm ammo.

I the movie Full Metal Jacket is to be believed, the 7.62mm round was also in use during the Vietnam War, but according to Wikipedia, the M16 also seemed to be using the 5.56mm round.

I was just wondering why they chose to go with a smaller or bigger diameter for their rounds. With the high availability of the AK-47 worldwide and therefore lots of 9mm producers it's probably not "we don't want that their bullets fit into our guns"